How to talk to aliens

In the 21st century, there is a great and serious enterprise afoot, of sending signals, as well as scanning all incoming signals for intelligent communication, in order to make contact with anyone out there in space. Guess what kind of signals the radio transmitters are sending, and the receivers are searching for! Yes, mathematics is the universal language, perhaps the only one that any form of intelligent life would quickly recognise and understand. How should we signal that we are here? Perhaps some sequences of blips, beginning with the simple and `natural', and ranging to the more complex - more obviously contrived by intelligence:

1,2,3,4,¼, - or blip, blip-blip, blip-blip-blip, etc. (the natural numbers).
Another way of communicating these numbers would be alternating two forms of electronic pulse, using binary notation:
*,*-,**,*-,*-*,**-,***,*-,¼ which we would write as:
1,10,11,100,101,110,111,1000,¼. Here are some sequences appropriate to signalling the presence of rather more intelligent beings:

1,3,5,7,9,11,¼ (the odd numbers)
1,8,15,22,29,36,¼ (the numbers equal to 1 (modulo  7))
1,3,6,10,15,21,¼ (the triangular numbers)
3,8,15,24,35,48,¼ (the numbers 1 less than a square)
2,9,28,65,126,217,¼ (the numbers 1 greater than a cube)
1,2,4,8,16,32,¼ (the exponentials 2n)
1,1,2,3,5,8,13,21,34,¼ (the Fibonacci numbers)
2,3,5,7,11,13,17,19,23,¼ (the prime numbers)
1,2,6,24,120,720, 5040,¼ (the factorials: n!)
1,2,2,3,2,4,2,4,3,4,2,6,¼ (the number of factors s(n))
1,6,28,496,8128,33550336,¼ (the perfect numbers)
This last one would be rather cumbersome in base 2, and unless our aliens have ten biological protuberances like our fingers they are unlikely to use base 10 normally. But perhaps they are intelligent enough to recognise any base! Here is a tricky one:
1,2,3,4,5,3,7,4,6,5,¼ (the Smarandache sequence)
It's defined by: S(n) is the smallest integer such that S(n)! is divisible by n.

Would you recognise the patterns? Could you translate them into base 2 for the transmitting engineers? This can be made into a game - Guess My Sequence, to add to the two games Guess My Number, and Guess My Rule, described by Richard Knottenbelt in Issue 5.1 (page 10). Each team is given a sequence expressed in binary form, either written on paper or (preferably) tapped out using two different sounds, with a third sound (or a silence) for the commas. The aim is to find the pattern and send the NEXT term of the sequence back in binary form. (See also our Readers' Problems 2 & 3 on page .)

A very useful tool for identifying such sequences from the first few terms is Encyclopaedia of Integer Sequences by N J A Sloane and Simon Plouffes, Academic Press 1995; on-line version at http://www.research.att.com/~ njas/sequences/Seis.html

And before there was radio?

Jules Verne's novel ``From the Earth to the Moon'' was written in the nineteenth century, long before humans actually visited the Moon; at that time telescopes, but not radio-communication, were in use. Imagine that you are living in that period; how would you signal evidence to the possible Moon-dwellers that we humans are here?

In the passage quoted below, the idea is put forward of communicating with alien intelligences, in the universal language of mathematics. But it has to be visual communication...

... a few days ago a German geometrician proposed to send a scientific expedition to the steppes of Siberia. There, on those vast plains, they were to describe enormous geometric figures, drawn in characters of reflecting luminosity, among which was the proposition regarding `the square of the hypotenuse', commonly called the `Ass's Bridge' by the French. `Every intelligent being,' said the geometrician, `must understand the scientific meaning of that figure. The Selenites, do they exist, will respond by a similar figure; and, a communication being thus once established, it will be easy to form an alphabet which shall enable us to converse with the inhabitants of the moon.' So spoke the German geometrician, but his project was never put into practice, and up to the present day there is no bond in existence between the earth and its satellite.

What was this geometric figure which every intelligent being must understand? (Actually, it's on our cover.)

It's the diagram for Euclid's Elements, Book 1, Proposition 27, where he proves Pythagoras' theorem! Here it is, on the left, as it might be drawn on the plains of Siberia. Well, are you an intelligent being? Then you cannot leave this page until you understand its meaning! There is a right-angled triangle, with the three squares drawn on its sides. That illustrates the proposition, that ``the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.''

The other lines illustrate the proof given by Euclid. Two large triangles are formed, drawn in bold in the diagram on the right, and the point is that they are congruent! This is because they have two sides (marked), and the included angle, equal (each angle is a right angle plus the same angle). Earlier propositions in Euclid guarantee that this is sufficient for triangles to be congruent.

But each triangle has half the area of the shaded square/rectangle on whose base it stands, because it has the same perpendicular height from this base. This is by another earlier proposition. Therefore the shaded areas are equal. Similarly, the other (unshaded) square is equal in area to the (unshaded) remainder of the big square. The result is proved! A triumph of deduction for the ancient Greeks, an inspiration for human reason ever since, and (according to Jules Verne) the common heritage of all intelligent beings!

WHAT'S THE MISTAKE? Jules Verne's geometry may have been alright, but his history was not! We offer a PRIZE for the best explanation of what Ass's Bridge meant, and where Mr Verne is wrong.

File translated from TEX by TTH, version 3.01.
On 28 Oct 2001, 10:41.